\Header{arima0}{ARIMA Modelling of Time Series -- Preliminary Version}
\alias{predict.arima0}{arima0}
\alias{print.arima0}{arima0}
\keyword{ts}{arima0}
\begin{Description}\relax
Fit an ARIMA model to a univariate time series, and forecast from
the fitted model.\end{Description}
\begin{Usage}
\begin{verbatim}
arima0(x, order = c(0, 0, 0),
       seasonal = list(order = c(0, 0, 0), period = NA),
       xreg = NULL, include.mean = TRUE, delta = 0.01,
       transform.pars = TRUE, fixed = NULL, init = NULL,
       method = c("ML", "CSS"), n.cond, optim.control = list())

## S3 method for class 'arima0':
predict(object, n.ahead = 1, newxreg, se.fit = TRUE, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a univariate time series
\item[\code{order}] A specification of the non-seasonal part of the ARIMA
model: the three components \eqn{(p, d, q)}{} are the AR order, the
degree of differencing, and the MA order.
\item[\code{seasonal}] A specification of the seasonal part of the ARIMA
model, plus the period (which defaults to \code{frequency(x)}).
This should be a list with components \code{order} and
\code{period}, but a specification of just a numeric vector of
length 3 will be turned into a suitable list with the specification
as the \code{order}.
\item[\code{xreg}] Optionally, a vector or matrix of external regressors,
which must have the same number of rows as \code{x}.
\item[\code{include.mean}] Should the ARIMA model include
a mean term? The default is \code{TRUE} for undifferenced series,
\code{FALSE} for differenced ones (where a mean would not affect
the fit nor predictions).
\item[\code{delta}] A value to indicate at which point \sQuote{fast
recursions} should be used.  See the Details section.
\item[\code{transform.pars}] Logical.  If true, the AR parameters are
transformed to ensure that they remain in the region of
stationarity.  Not used for \code{method = "CSS"}.
\item[\code{fixed}] optional numeric vector of the same length as the total
number of parameters.  If supplied, only \code{NA} entries in
\code{fixed} will be varied.  \code{transform.pars = TRUE}
will be overridden (with a warning) if any ARMA parameters are
fixed.
\item[\code{init}] optional numeric vector of initial parameter
values. Missing values will be filled in, by zeroes except for
regression coefficients.
\item[\code{method}] Fitting method: maximum likelihood or minimize
conditional sum-of-squares.
\item[\code{n.cond}] Only used if fitting by conditional-sum-of-squares: the
number of initial observations to ignore.  It will be ignored if
less than the maximum lag of an AR term.
\item[\code{optim.control}] List of control parameters for \code{\Link{optim}}.
\item[\code{object}] The result of an \code{arima0} fit.
\item[\code{newxreg}] New values of \code{xreg} to be used for
prediction. Must have at least \code{n.ahead} rows.
\item[\code{n.ahead}] The number of steps ahead for which prediction is required.
\item[\code{se.fit}] Logical: should standard errors of prediction be returned?
\item[\code{...}] arguments passed to or from other methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Different definitions of ARMA models have different signs for the
AR and/or MA coefficients. The definition here has

\deqn{X_t = a_1X_{t-1} + \cdots + a_pX_{t-p} + e_t + b_1e_{t-1} + \dots + b_qe_{t-q}}{\code{X[t] = a[1]X[t-1] + ... + a[p]X[t-p] + e[t] + b[1]e[t-1] + ... + b[q]e[t-q]}}

and so the MA coefficients differ in sign from those of
S-PLUS.  Further, if \code{include.mean} is true, this formula
applies to \eqn{X-m}{} rather than \eqn{X}{}.  For ARIMA models with
differencing, the differenced series follows a zero-mean ARMA model.

The variance matrix of the estimates is found from the Hessian of
the log-likelihood, and so may only be a rough guide, especially for
fits close to the boundary of invertibility.

Optimization is done by \code{\Link{optim}}. It will work
best if the columns in \code{xreg} are roughly scaled to zero mean
and unit variance, but does attempt to estimate suitable scalings.

Finite-history prediction is used. This is only statistically
efficient if the MA part of the fit is invertible, so
\code{predict.arima0} will give a warning for non-invertible MA
models.\end{Details}
\begin{Value}
For \code{arima0}, a list of class \code{"arima0"} with components:

\begin{ldescription}
\item[\code{coef}] a vector of AR, MA and regression coefficients,
\item[\code{sigma2}] the MLE of the innovations variance.
\item[\code{var.coef}] the estimated variance matrix of the coefficients
\code{coef}.
\item[\code{loglik}] the maximized log-likelihood (of the differenced data),
or the approximation to it used.
\item[\code{arma}] A compact form of the specification, as a vector giving
the number of AR, MA, seasonal AR and seasonal MA coefficients,
plus the period and the number of non-seasonal and seasonal
differences.
\item[\code{aic}] the AIC value corresponding to the log-likelihood. Only
valid for \code{method = "ML"} fits.  Prior to \R{} 1.5.0 this omitted
\code{2} for the estimation of the innovations variance.
\item[\code{residuals}] the standardized residuals. (Was \code{resid}
prior to \R{} 1.5.0.)
\item[\code{call}] the matched call.
\item[\code{series}] the name of the series \code{x}.
\item[\code{convergence}] the value returned by \code{\Link{optim}}.
\item[\code{n.cond}] the number of initial observations not used in the fitting.
\end{ldescription}


For \code{predict.arima0}, a time series of predictions, or if
\code{se.fit = TRUE}, a list with components \code{pred}, the
predictions, and \code{se}, the estimated standard errors. Both
components are time series.\end{Value}
\begin{Section}{Fitting methods}
The exact likelihood is computed via a state-space representation of
the ARMA process, and the innovations and their variance found by a
Kalman filter based on Gardner \emph{et al.} (1980).  This has
the option to switch to \sQuote{fast recursions} (assume an
effectively infinite past) if the innovations variance is close
enough to its asymptotic bound. The argument \code{delta} sets the
tolerance: at its default value the approximation is normally
negligible and the speed-up considerable.  Exact computations can be
ensured by setting \code{delta} to a negative value.

If \code{transform.pars} is true, the optimization is done using an
alternative parametrization which is a variation on that suggested by
Jones (1980) and ensures that the model is stationary.  For an AR(p)
model the parametrization is via the inverse tanh of the partial
autocorrelations: the same procedure is applied (separately) to the
AR and seasonal AR terms.  The MA terms are also constrained to be
invertible during optimization by the same transformation if
\code{transform.pars} is true.  Note that the MLE for MA terms does
sometimes occur for MA polynomials with unit roots: such models can be
fitted by using \code{transform.pars = FALSE} and specifying a good
set of initial values (often obtainable from a fit with
\code{transform.pars = TRUE}).

As from \R{} 1.5.0 missing values are allowed, but any missing values
will force \code{delta} to be ignored and full recursions used.
Note that missing values will be propogated by differencing, so the
procedure used in this function is not fully efficient in that case.

Conditional sum-of-squares is provided mainly for expositional
purposes.  This computes the sum of squares of the fitted innovations
from observation
\code{n.cond} on, (where \code{n.cond} is at least the maximum lag of
an AR term), treating all earlier innovations to be zero.  Argument
\code{n.cond} can be used to allow comparability between different
fits.  The \dQuote{part log-likelihood} is the first term, half the
log of the estimated mean square.  Missing values are allowed, but
will cause many of the innovations to be missing.

When regressors are specified, they are orthogonalized prior to
fitting unless any of the coefficients is fixed.  It can be helpful to
roughly scale the regressors to zero mean and unit variance.\end{Section}
\begin{Note}\relax
This is a preliminary version, and will be replaced by \code{\Link{arima}}.

The standard errors of prediction exclude the uncertainty in the
estimation of the ARMA model and the regression coefficients.

The results are likely to be different from S-PLUS's
\code{arima.mle}, which computes a conditional likelihood and does
not include a mean in the model.  Further, the convention used by
\code{arima.mle} reverses the signs of the MA coefficients.\end{Note}
\begin{References}\relax
Brockwell, P. J. and Davis, R. A. (1996) \emph{Introduction to Time
Series and Forecasting.} Springer, New York. Sections 3.3 and 8.3.

Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980) Algorithm
AS154. An algorithm for exact maximum likelihood estimation of
autoregressive-moving average models by means of Kalman filtering.
\emph{Applied Statistics} \bold{29}, 311--322.

Harvey, A. C. (1993) \emph{Time Series Models},
2nd Edition, Harvester Wheatsheaf, sections 3.3 and  4.4.

Harvey, A. C. and McKenzie, C. R. (1982) Algorithm AS182.
An algorithm for finite sample prediction from ARIMA processes.
\emph{Applied Statistics} \bold{31}, 180--187.

Jones, R. H. (1980) Maximum likelihood fitting of ARMA models to time
series with missing observations. \emph{Technometrics} \bold{20} 389--395.\end{References}
\begin{SeeAlso}\relax
\code{\Link{arima}}, \code{\Link{ar}}, \code{\Link{tsdiag}}\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(lh)
## Don't run: arima0(lh, order = c(1,0,0))
arima0(lh, order = c(3,0,0))
arima0(lh, order = c(1,0,1))
predict(arima0(lh, order = c(3,0,0)), n.ahead = 12)

arima0(lh, order = c(3,0,0), method = "CSS")

data(USAccDeaths)
# for a model with as few years as this, we want full ML
(fit <- arima0(USAccDeaths, order = c(0,1,1),
               seasonal = list(order=c(0,1,1)), delta = -1))
predict(fit, n.ahead = 6)

data(LakeHuron)
arima0(LakeHuron, order = c(2,0,0), xreg = time(LakeHuron)-1920)
## Don't run: 
data(presidents) # contains NAs
## graphs in example(acf) suggest order 1 or 3
(fit1 <- arima0(presidents, c(1, 0, 0), delta = -1))  # avoid warning
tsdiag(fit1)
(fit3 <- arima0(presidents, c(3, 0, 0), delta = -1))  # smaller AIC
tsdiag(fit3)
## End Don't run
\end{ExampleCode}
\end{Examples}

